The word “casino” comes from the Italian. It's a diminutive of the common Romantic-language term casa, meaning house. So it means little house (or country villa).
In 16th–17th century Italy, a casino was a small social building or villa used for gatherings, music, dancing, and conversation — not gambling. Within 100 years or so, these social houses started hosting gambling games. The most notable was Casino di Venezia, one of the earliest casinos with gambling. (You can read our Question of the Day about that here.) Over the centuries, the original meaning of the word was supplanted by the new meaning: public establishments for gambling, often combined with entertainment, dining, and lodging.
But the Etymological evolution of casino didn't stop there. Today, in Italian, it can also mean “a mess” or “chaos” — which many gamblers might find oddly appropriate.
We've been told by Italian speakers it can also mean brothel, in the sense that a "little house” might be used for private or illicit activities, and in 18th–19th century Italy, casini (the plural of casino in Italian) were sometimes rented for discreet romantic encounters.
However, these days, we're told it's decidedly colloquial and context-dependent and much less common than its other meanings, especially a mess or chaos. And we're sure you'll neglect to add this last bit of information when you relate the origins of "casino" to your nine-year-old daughter.